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Full text of "Treatise On Analysis Vol-Ii"

8    LEBESGUE'S CONVERGENCE THEOREMS       129

from which it follows ((3.16.5) and (8.7.8)) that, for almost all x, the function

f(x, z) is the limit of the sequence I/T1 Y,fkn(x)}9 where

\     fc-i        /

_/(*, a + re2kni/n)re2kni/n

Jkn\XJ ""               !------2kni/n-----------"" *

a + re2kni/n  z
Applying (13.8.4), it follows that h(z) is the limit of the sequence

and therefore that

which shows that h is analytic in E (9.9.4). By replacing the term !/((  z) in
the right-hand side of (13.8.6.2) by fc!/(f-z)*+1, we obtain the formula
stated for D*/z(z).

(13.8.7)   (i)   If (An) is a sequence of integrable sets, then A = () An w

n

integrable. If also the sequence (Aw) w decreasing, we have
(13.8.7.1)                                MA) = mfXAJ.

n

(ii)   If(An) is a sequence of integrable sets contained in an integrable set B,
then A = (J AM w integrable.

n

(iii)   If(An) is a sequence of integrable sets such that

11=1

/on A = (J Art is integrable, and we have

(13.8.7.2)

=1

If moreover the sets An are pairwise disjoint, then
(13.8.7.3)                               /i(A)=

(" complete additivity of measure *'). the right-hand side of (13.8.6.1). But it follows from the hypotheses